Functions

Video lesson5 worked examples

Theory

A function is a mathematical relationship often denoted using functional notation like $f(x)$ or $g(x)$ .

You must be able to substitute values into a function to evaluate it, and determine a missing variable when a function's overall output is given.

The Golden Rule: $f(\text{something})$ means “replace every $x$ in the formula with that something”. Put the value in brackets when you substitute — it keeps signs and powers correct.

⚠️ Common Examiner Traps

Squaring a negative: $f(-3)$ with an $x^2$ term gives $(-3)^2 = 9$ , positive — not $-9$ . Always bracket the substituted value.
$f(x) = 0$ is not the same as $f(0)$ : the first solves an equation for $x$ ; the second substitutes $x = 0$ .
Reading the notation backwards: in $g(a) = 35$ , the $35$ is the output — set the formula equal to it and solve for $a$ .

Worked examples

Example 1

Evaluating a function

A function is defined by $f(x) = x^2 - 5x$ . Calculate $f(-3)$ .

Step 1: Substitute −3 in place of x: $(-3)^2 - 5(-3)$ .

Step 2: Calculate: $9 + 15$ .

Answer: 24.

Example 2

Finding an unknown

A function is defined as $g(t) = 4t + 7$ . Given that $g(a) = 35$ , find a.

Step 1: Set the expression equal to 35: $4a + 7 = 35$ .

Step 2: Solve for a: $4a = 28 \rightarrow a = 7$ .

Example 3

Finding an unknown (fractional function)

A function is defined by $f(x) = 2 + \dfrac{6}{x}$ . Given that $f(a) = 5$ , find $a$ .

Step 1: Substitute and set equal to 5: $2 + \dfrac{6}{a} = 5$ .

Step 2: Subtract 2 from both sides: $\dfrac{6}{a} = 3$ .

Step 3: Multiply both sides by $a$ , then divide by 3: $6 = 3a \rightarrow a = 2$ .

Answer: $a = 2$ .

Example 4

Evaluating a non-polynomial function

A function is defined by $f(x) = 10\sin x^\circ$ . Evaluate $f(30)$ .

Step 1: Function notation works the same whatever the formula — substitute $30$ for $x$ : $f(30) = 10\sin 30^\circ$ .

Step 2: Use the exact value $\sin 30^\circ = \tfrac{1}{2}$ : $10 \times \tfrac{1}{2}$ .

Answer: 5.

Example 5

🔗 Bringing it together

A function is defined by $f(x) = x^2 - 5x$ . Find the values of $x$ for which $f(x) = 0$ .

Step 1: $f(x) = 0$ means set the formula equal to zero — an equation to solve, not a substitution: $x^2 - 5x = 0$ .

Step 2: Factorise (common factor of $x$ ): $x(x - 5) = 0$ .

Step 3: A product is zero when either factor is zero, so $x = 0$ or $x - 5 = 0$ .

Answer: $x = 0$ or $x = 5$ .

Ready to practise?

16 Functions questions with answers and video solutions